About the theory of local homology and local cohomology modules

We prove some results concerning minimaxness and finiteness of local homology modules and by Matlis duality we extend some results for the minimaxness and finiteness of local cohomology modules. We introduce the concept of C-minimax R-modules, and we discuss the maximum and minimum integers such that local homology and local cohomology modules are C-minimax. As a consequence, we find minimum integers such that local homology and local cohomology modules are of finite length.

We show that if M is a finitely generated R-module, N is an I-stable semi-discrete linearly compact R-module and G is a closed R-submodule of the generalized local homology module such that is I-stable for all j < i and is I-stable, then the set Coass G is finite. As an immediate consequence, the first non-I-stable generalized local homology module of N, M with respect to I has only finitely many co-associated primes. By duality, we get a new result for the finiteness of associated primes of (generalized) local cohomology modules.

In this paper, we obtain some results about the local homology modules of Artinian modules, and by Matlis duality we obtain some results about the local cohomology modules of finitely generated modules.

We study some properties of representable or I-stable local homology modules HiI (M) where M is a linearly compact module. By duality, we get some properties of good or at local cohomology modules HIi (M) of A. Grothendieck.

We show some results about local homology modules and local cohomology modules concerning Grothendieck’s conjecture and Huneke’s question. We also show some equivalent properties of [Formula: see text]-separated modules and of minimax local homology modules. By duality, we get some properties of Grothendieck’s local cohomology modules.

A local homology theory for linearly compact modules

We show some results about local homology modules and local cohomology modules concerning to being in a serre sub category of the category of R-modules. Also for an ideal I of R we define the concept of CI condition on a serre category, which seems dual to CI condition of Melkerson [1]. As a main result we show that for any minimax R-module M of any serre category S that satisifies CI (CI) condition the local homology module HiI(M) (local cohomology module HIi(M) 2 S) for all i ≥ 0.

Coassociated primes of local homology and local cohomology modules

In this article, we can to relate the theory of local cohomology, with respect to an ideal, to the theory of local homology modules. With the results of the article, we show the importance of local homology theory as a study tool within of the commutative algebra theory.

The first part of the paper is concerned to relationship between the sets of associated primes of the generalized $d$-local cohomology modules and the ordinary generalized local cohomology modules. Assume that $R$ is a commutative Noetherian local ring, $M$ and $N$ are finitely generated $R$-modules and $d, t$ are two integers. We prove that $Ass H^t_d(M,N)=bigcup_{Iin Phi} Ass H^t_I(M,N)$ whenever $H^i_d(M,N)=0$ for all $i< t$ and $Phi={I: I text{ is an ideal of} R text{ with} dim R/Ileq d }$. In the second part of the paper, we give some information about the non-vanishing of the generalized $d$-local cohomology modules. To be more precise, we prove that $H^i_d(M,R)neq 0$ if and only if $i=n-d$ whenever $R$ is a Gorenstein ring of dimension $n$ and $pd_R(M)<infty$. This result leads to an example which shows that $Ass H^{n-d}_d(M,R)$ is not necessarily a finite set.

Constructing arbitrary torsion elements for a local cohomology module

where K.(x1 , . . . , x n r , M) is the Koszul complex of M with respect to x n 1 , . . . , x n r . Then H p (·) is an additive, A-linear covariant functor from the category of Artinian A-modules and A-homomorphisms to the category of A-modules and Ahomomorphisms and there is a long exact sequence of local homology modules for any short exact sequence of Artinian A-modules. For any ideal I of A, by [4,Lemma 3], there exist finitely many elements x1, . . . , xr ∈ I such that 0 :M I = 0 :M (x1, . . . , xr). It is proved that H x p (M) does not depend on the choice of x = (x1, . . . , xr), up to A-isomorphisms. We use H I p (M) to denote one of these H p (M). The two important concepts of Artinian modules, cograde and dimension, are connected with local homology modules. The paper is carried out when the author is visiting the Department of Pure Mathematics of Sheffield University. I am grateful to Prof.R.Y.Sharp for helpful discussions.

Let $A$ be a regular local ring of positive characteristic. This paper is concerned with the local cohomology modules of $A$ itself, but with respect to an arbitrary ideal of $A$. The results include that all the Bass numbers of all such local cohomology modules are finite, that each such local cohomology module has finite set of associated prime ideals, and that, whenever such a local cohomology module is Artinian, then it must be injective. (This last result had been proved earlier by Hartshorne and Speiser under the additional assumptions that $A$ is complete and contains its residue field which is perfect.) The paper ends with some low-dimensional evidence related to questions about whether the analogous statements for regular local rings of characteristic $0$ are true.

Let [Formula: see text] be ideals of a commutative Noetherian ring [Formula: see text] and [Formula: see text] be a finitely generated [Formula: see text]-module. By using filter regular sequences, we show that the infimum of integers [Formula: see text] such that the local cohomology modules [Formula: see text] and [Formula: see text] are not isomorphic is equal to the infimum of the depths of [Formula: see text]-modules [Formula: see text], where [Formula: see text] runs over all prime ideals of [Formula: see text] containing only one of the ideals [Formula: see text]. In particular, these local cohomology modules are isomorphic for all integers [Formula: see text] if and only if [Formula: see text]. As an application of this result, we prove that for a positive integer [Formula: see text], [Formula: see text] is Artinian for all [Formula: see text] if and only if, it can be represented as a finite direct sum of [Formula: see text] local cohomology modules of [Formula: see text] with respect to some maximal ideals in [Formula: see text] for any [Formula: see text]. These representations are unique when they are minimal with respect to inclusion.

In this paper we study relative and Tate cohomology of modules of finite Gorenstein injective dimension. Using these cohomology theories, we present variations of Grothendieck local cohomology modules, namely Gorenstein and Tate local cohomology modules. By applying a sort of Avramov-Martsinkovsky exact sequence, we show that these two variations of local cohomology are tightly connected to the generalized local cohomology modules introduced by J. Herzog. We discuss some properties of these modules and give some results concerning their vanishing and non-vanishing.